1549380323-Statistical Mechanics Theory and Molecular Simulation

466 The Feynman path integral

∫t

0

ds

[

m

2

(

dx

ds

) 2

−U(x(s))

]

=

∫−iβh ̄

0

d(−iτ)

[

m

2

(

dx

d(−iτ)

) 2

−U(x(−iτ))

]

=i

∫β ̄h

0

dτ

[

m

2

(

dx

dτ

) 2

+U(x(τ))

]

≡iS[x]. (12.4.11)

Note that the actionS[x] is now the action for paths inimaginary timeτthat start at
x(0) =xand end atx(τ) =x′in imaginary timeτ=β ̄h. The imaginary-time action
S[x] differs from the real-time actionA[x] by the sign of the potential. The actionS[x]
is often called theEuclidean action. In terms of imaginary-time paths, we may write
the density matrix elements as

ρ(x,x′;β) =

∫x(β ̄h)=x′

x(0)=x

Dx(τ) exp

{

−

1

̄h

∫βh ̄

0

dτ

[

1

2

mx ̇^2 (τ) +U(x(τ))

]}

=

∫x(β ̄h)=x′

x(0)=x

Dx(τ) exp

{

−

1

̄h

∫βh ̄

0

dτΛ(x(τ),x ̇(τ))

}

=

∫x′

x

Dxe−S[x]/ ̄h. (12.4.12)

The quantity Λ(x,x ̇) = (m/2) ̇x^2 +U(x) is called theimaginary-time Lagrangianor
Euclidean Lagrangian. The density matrix is constructed by integrating over all paths
x(τ) that satisfyx(0) =x,x(β ̄h) =x′weighted by exp(−S[x]/ ̄h). Since this weight
factor is positive definite, we can find the most important contributions to the func-
tional integral in eqn. (12.4.12) by minimizing the Euclidean action with respect to
the pathx(τ). As we did for the propagator, we consider a pathx(τ) and a nearby
path ̃x(τ) =x(τ) +δx(τ). Ifx(0) =xandx(β ̄h) =x′, thenδx(0) =δx(β ̄h) = 0.
We require that the variationδS=S[x+δx]−S[x] vanish to first order in the path
variationδx. Following the procedure in Section 1.8, the resulting equation of motion
will be exactly the form of the Euler–Lagrange equation applied to Λ:

d

dτ

(

∂Λ

∂x ̇(τ)

)

−

∂Λ

∂x(τ)

= 0. (12.4.13)

However, when we apply the Euler–Lagrange equation to the Euclidean Lagrangian,
we obtain an equation of motion of the form

m

d^2 x

dτ^2

=

∂U

∂x

. (12.4.14)

Eqn. (12.4.14) resembles Newton’s second law except that the force is calculated using
not the potentialU(x) but an inverted potential surface−U(x). This result is not
unexpected: If we transform Newton’s second law in real timemd^2 x/ds^2 =−∂U/∂x